Applied mathematics

Efficient solutions to the vehicle routing problem require tools from combinatorial optimization and integer programming. Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems; as a profession focused on practical problems, applied mathematics focuses on the formulation and study of mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics. Contents 1 Divisions 2 Utility 3 Status in academic departments 4 Other associated mathematical sciences 4.1 Scientific computing 4.2 Computer Science 4.3 Operations research and management science 4.4 Statistics 4.5 Actuarial science 4.6 Other disciplines 5 See also 6 References 7 External links // Divisions A numerical solution to the heat equation on a pump casing model using the finite element method. There is no consensus as to what the various branches of applied mathematics are. Such categorizations are made difficult by the way mathematics and science change over time, and also by the way universities organize departments, courses, and degrees. Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variational methods, and numerical analysis); and applied probability. These areas of mathematics were intimately tied to the development of Newtonian physics, and in fact the distinction between mathematicians and physicists was not sharply drawn before the mid-19th century. This history left a legacy as well: until the early 20th century subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments, and fluid mechanics may still be taught in applied mathematics departments.1 As well as physics, engineering and computer science have traditionally made use of applied mathematics. Fluid mechanics is often considered a branch of applied mathematics. Today, the term applied mathematics is used in a broader sense. It includes the classical areas above, as well as other areas that have become increasingly important in applications. Even fields such as number theory that are part of pure mathematics are now important in applications (such as cryptography), though they are not generally considered to be part of the field of applied mathematics per se. Sometimes the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today. Many mathematicians distinguish between applied mathematics, which is concerned with mathematical methods, and the applications of mathematics within science and engineering. A biologist using a population model and applying known mathematics would not be doing applied mathematics, but rather using it; however mathematical biologists have posed problems that have stimulated the growth of pure mathematics. However, mathematicians like Poincaré and Arnold deny the existence of "applied mathematics" and claim that there are only "applications of mathematics"; similarly, nonmathematicians blend applied mathematics and applications of mathematics. The use and development of mathematics to solve industrial problems is also called industrial mathematics.2 The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics, computational science, and computational engineering, which use high performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering. These are often considered interdisciplinary disciplines. Utility Mathematical finance is concerned with the modelling of financial markets. Historically, mathematics was most important in the natural sciences and engineering. However, since World War II, fields outside of the physical sciences have spawned the creation of new areas of mathematics, such as game theory and social choice theory, which grew out of economic considerations, or neural networks, which arose out of the study of the brain in neuroscience. The advent of the computer has created new applications: studying and using the new computer technology itself (computer science), using computers to study problems arising in other areas of science (computational science), and studying the mathematics of computation (numerical analysis). Statistics is probably the most widespread mathematical science used in the social sciences, but other areas of mathematics are proving increasingly useful in these disciplines, most notably in economics. Status in academic departments This section does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2008) Academic institutions are not consistent in the way they group and label courses, programs, and degrees in applied mathematics. At some schools, there is a single mathematics department, whereas others have separate departments for Applied Mathematics and (Pure) Mathematics. It is very common for Statistics departments to be separate at schools with graduate programs, but many undergraduate-only institutions include statistics under the mathematics department. Many applied mathematics programs (as opposed to departments) consist of primarily cross-listed courses and jointly-appointed faculty in departments representing applications. Some Ph.D. programs in applied mathematics require little or no coursework outside of mathematics, while others require substantial coursework in a specific area of application. In some respects this difference reflects the distinction between "application of mathematics" and "applied mathematics". Some universities in the UK host departments of Applied Mathematics and Theoretical Physics, but it is now much less common to have separate departments of pure and applied mathematics. A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, housing the Lucasian Professor of Mathematics whose past holders include Isaac Newton, Charles Babbage, James Lighthill, Paul Dirac and Stephen Hawking. Schools with separate applied mathematics departments range from Brown University, which has a well-known and large Division of Applied Mathematics that offers degrees through the doctorate, to Santa Clara University, which offers only the M.S. in applied mathematicscitation needed. Research universities dividing their mathematics department into pure and applied sections include Harvard and MIT. At some universities there is some tension between applied and pure mathematics departments. One reason is that pure mathematics is often perceived as having a higher intellectual standing. Another reason is a different level of compensation, as applied mathematicians are often paid more. Applied mathematics also enjoys better opportunities to bring external funding from many sources, not limited to the Division of Mathematical Sciences at the National Science Foundation (NSF) like much of pure mathematics. External funding is highly valued at research universities and is often a condition for faculty advancement. Similar tensions can also exist between statistics and mathematics groups and departments. Other associated mathematical sciences Applied mathematics has substantial overlap with statistics. Applied mathematics is closely related to other mathematical sciences. Scientific computing Scientific computing includes applied mathematics (especially numerical analysis), computing science (especially high-performance computing), and mathematical modelling in a scientific discipline. Computer Science Computer science relies on logic, algebra, and combinatorics. Operations research and management science Operations research and management science are often taught in faculties of engineering, business, public policy. Statistics Applied mathematics has substantial overlap with the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Statistical theory relies on probability and decision theory, and makes extensive use of scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra and combinatorial design. Applied mathematicians and statisticians often work in a department of mathematical sciences (particularly at colleges and small universities). Actuarial science Actuarial science uses probability, statistics, and economic theory. Other disciplines The line between applied mathematics and specific areas of application is often blurred. Many universities teach mathematical and statistical courses outside of the respective departments, in departments and areas including business and economics, engineering, physics, chemistry, psychology, biology, computer science, and mathematical physics. See also Mathematics portal Book: Applied science Wikipedia Books are collections of articles that can be downloaded or ordered in print. Society for Industrial and Applied Mathematics References ^ Stolz, M. (2002), "The History Of Applied Mathematics And The History Of Society", Synthese 133 (1): 43–57, doi:10.1023/A:1020823608217, http://www.springerlink.com/index/T152575218M865W4.pdf, retrieved 2009-07-07  ^ University of Strathclyde (17 January 2008), Industrial Mathematics, http://www.maths.strath.ac.uk/applying/postgraduate/research_topics/industrial_mathematics, retrieved 8 January 2009  External links Wikiversity has learning materials about School of Mathematics The Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to promoting the interaction between mathematics and other scientific and technical communities. Aside from organizing and sponsoring numerous conferences, SIAM is a major publisher of research journals and books in applied mathematics. v · d · eTopics in industrial & applied mathematics Computational mathematics Computational logic • Algorithm design & analysis • Information theory • Coding theory • Cryptography Discrete mathematics Computer algebra • Computational number theory • Combinatorics • Graph theory • Discrete geometry Algebraic physics Operator algebra • Representation theory • Renormalization group • Feynman integral • M-theory Analytic physics Differential geometry • Fourier analysis • Harmonic analysis •Functional analysis • Operator theory Analysis Approximation theory • Numerical analysis • Differential equations • Dynamical systems • Control theory • Variational calculus Probability theory Distributions of random variables • Stochastic process • Stochastic analysis • Path integral • Stochastic variational calculus Decision sciences Statistics • Operations research • Optimization • Game theory •Mathematical economics • Mathematical finance Other applications Applicability of mathematics (philosophy) • Biology • Chemistry • Psychology • Sociology Category  · Mathematics portal  · Outline  · Topics v · d · eAreas of mathematics Areas Arithmetic · Algebra (elementary – linear – abstract) · Geometry (Discrete geometry – Algebraic geometry – Differential geometry) · Calculus/Analysis · Set theory · Logic · Category theory · Number theory · Combinatorics · Graph theory · Topology · Lie theory · Differential equations/Dynamical systems · Mathematical physics · Numerical analysis · Computation · Information theory · Probability · Statistics · Optimization · Control theory · Game theory Large divisions Pure mathematics · Applied mathematics · Discrete mathematics · Computational mathematics Category  · Mathematics portal  · Outline · Lists